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Published for SISSA by SpringerReceived: July 17, 2014 Accepted: September 9, 2014 Published: October 3, 2014
Bounds on OPE coefficients in 4D Conformal Field
Theories
Francesco Caracciolo,a Alejandro Castedo Echeverri,a Benedict von Harlinga and Marco Seronea,b
aSISSA and INFN,
Via Bonomea 265, I-34136 Trieste, Italy
bICTP,
Strada Costiera 11, I-34151 Trieste, Italy
E-mail: acastedo@sissa.it,bharling@sissa.it,serone@sissa.it
Abstract: We numerically study the crossing symmetry constraints in 4D CFTs, using previously introduced algorithms based on semidefinite programming. We study bounds on OPE coefficients of tensor operators as a function of their scaling dimension and extend previous studies of bounds on OPE coefficients of conserved vector currents to the prod-uct groups SO(N )×SO(M). We also analyze the bounds on the OPE coefficients of the conserved vector currents associated with the groups SO(N ), SU(N ) and SO(N )×SO(M) under the assumption that in the singlet channel no scalar operator has dimension less than four, namely that the CFT has no relevant deformations. This is motivated by applications in the context of composite Higgs models, where the strongly coupled sector is assumed to be a spontaneously broken CFT with a global symmetry.
Keywords: Conformal and W Symmetry, Technicolor and Composite Models
ArXiv ePrint: 1406.7845
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Contents
1 Introduction 1
2 Motivation for a gap in the scalar operator dimension 3
3 Review of the bootstrap program 6
3.1 Bootstrap equations 6
3.2 Bounds on OPE coefficients and numerical implementation 8
4 Bounds on OPE coefficients for tensor operators 10
5 Bounds on current-current two-point functions 12
5.1 SO(N ) global symmetry 13
5.2 SU(N ) global symmetry 15
5.3 G1× G2 global symmetries 17
5.3.1 SO(N )×SO(M) 18
5.3.2 SO(N )×SU(M) 19
6 Conclusions 21
A Details about the numerical procedure 22
B Crossing relations for SO(N )×SO(M ) and SO(N )×SU(M ) 24
B.1 SO(N )×SO(M) 24
B.2 SO(N )×SU(M) 25
1 Introduction
There has recently been a renewed interest in studying general properties of four-dimensional (4D) Conformal Field Theories (CFTs) after the seminal paper [1] revived the bootstrap program advocated in the early 70s [2, 3]. Imposing the associativity of the Operator Product Expansion (OPE) and unitarity, ref. [1] has shown how one can set bounds on scalar operator dimensions in 4D CFTs. Although these constraints are based on numerical methods, they come from first principles, with no further assumptions. Since then, various generalizations of this result have been developed in order to improve the above bounds, to put bounds on OPE coefficients and on CFT data in presence of a global symmetry [4–10]. In particular, bounds were derived on the OPE coefficients associated with the energy momentum tensor (the central charge) and the conserved vector current of a global symmetry [6–10]. Superconformal field theories and CFTs in d6= 4 have also
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been considered [6,10–22]. In addition to the above mostly numerical results, considerable progress has also been made on more analytic aspects of CFTs, see e.g. refs. [23–36].
The aim of this paper is to numerically study the bounds on the coefficient κ of the two-point function between two conserved currents associated with a global symmetry of a CFT. Our main motivation comes from theoretical considerations in the context of composite Higgs models, in which the CFT is the hidden sector which gives rise to the Higgs, and a subgroup of the global symmetry of the CFT is weakly gauged in order to get the Standard Model gauge interactions. These composite Higgs models models are related, through the AdS/CFT correspondence, to Randall-Sundrum theories [37,38] with matter in the bulk, which are a promising solution to the gauge hierarchy problem. Particularly interesting are the models where the Higgs is a pseudo Nambu-Goldstone Boson (pNGB) of an approximate spontaneously broken global symmetry of the CFT, which correspond to gauge-Higgs unification models in 5D warped theories. Neither the UV completion of the 5D models nor the explicit form of the 4D CFT is known so far. Calculability of the dual 5D models would require that the CFT is in some large N limit, but this is not a necessary requirement. On the contrary, various phenomenological bounds tend to favour models at small N, so we will not assume the existence of a large N limit in the CFT. Constructing such a CFT is not a trivial task, so we look for possible consistency relations. When the global symmetry of the CFT is gauged, the coefficient κ of the current-current two-point function governs the leading contribution of the CFT to the one-loop evolution of the corresponding gauge coupling. This contribution should not lead to Landau poles for the SM gauge couplings. We also require that the CFT has no relevant deformations, in order not to reintroduce the hierarchy problem. This leads to the constraint that the dimension of the lowest-lying scalar singlet operator should be ∆S ≥ 4. All our considerations apply
independently of the pNGB nature or not of the Higgs.
Motivated by the above considerations, we extend the analysis of ref. [10], where lower bounds on κ have been set starting from crossing constraints imposed on a four-point function of scalar operators in the fundamental representation of SO(N ) and SU(N ), in two ways. First, we see how the bounds found in ref. [10] are modified when the lowest-lying singlet scalar operator is assumed to have a scaling dimension ∆S ≥ ∆min, where
we choose ∆min = 2, 3, 4 for concreteness. Second, we extend the analysis to non-simple
groups of the form SO(N )×SO(M). We study non-simple groups because they easily allow to generalize the bounds for the groups SO(N ) and SU(N ), which are obtained by considering a single field in the fundamental representation of the group, to multiple fields. Analogous to what was found in ref. [10] for singlet operators, the lower bounds on vector currents for SU(N ) groups that we find are, within the numerical precision, identical to those obtained for SO(2N ). Hence we only report lower bounds for SO(N ) and SO(N )×SO(M) global symmetries. We have derived the bootstrap equations also for groups of the form SO(N )×SU(M), but no bounds are reported for this case, since we have numerical evidence that the lower bounds for SO(N )×SU(M) are essentially identical to those obtained for SO(N )×SO(2M), similarly to the above equality between SO(2N) and SU(N ) bounds.In addition to that, we study the constraints on the OPE coefficients of spin l = 2 and l = 4 tensors coming from two identical scalar operators φ, as a function
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of the scaling dimension of the tensors, in the general case in which no global symmetry is assumed. In analogy to the vector-current case, we analyze how these bounds change when one assumes a lower bound on the scale dimension of the scalar operators appearing in the φφ OPE.
All our numerical results are based on semi-definite programming methods, as intro-duced in ref. [10] in the context of the bootstrap approach, with a few technical modifica-tions which are discussed in subsection 3.2and in appendix A.
The structure of the paper is as follows. In section 2 we describe the phenomenological motivations behind our work. In section 3 we briefly review the basic properties of the crossing constraints coming from four-point functions of identical scalars and review how bounds on OPE coefficients are numerically obtained. In section 4 we report our results for the OPE coefficients of tensor l = 2 and l = 4 operators. Section 5 contains the most important results of the paper. We report here the lower bounds on κ associated with SO(2N ) (or SU(N )) vector currents, when the global symmetry of the CFT is SO(2N ) (or SU(N )) and SO(2N )× SO(M).1 In section 6 we conclude. Two appendices complete the paper. In appendix A we discuss various technical details about our implementation of the bootstrap equations in the semi-definite programming method, while in appendix B we report the crossing equations for SO(N )×SO(M) and SO(N)×SU(M).
2 Motivation for a gap in the scalar operator dimension
The motivation to consider CFTs with a gap in the scaling dimension of scalar gauge-singlet operators comes from applications in the context of physics beyond the Standard Model (SM) that addresses the gauge hierarchy problem. The latter can be formulated from a CFT point of view, see e.g. ref. [1]. Neglecting the cosmological constant, the SM can be seen as an approximate CFT with one relevant deformation of classical mass dimension ∆H†H = 2, corresponding to the Higgs mass term H†H. Relevant deformations grow in going from the UV towards the IR. If we assume that the Higgs mass term is generated at some high scale ΛU V, we would expect from naturalness that the Higgs mass-squared
term is of order Λ4−∆H†H
U V = Λ2U V in the IR. There are essentially two ways to solve this
hierarchy problem: i) invoke additional symmetries that keep the relevant deformation small in the IR (e.g. supersymmetry); ii) assume that the Higgs is a composite field of a strongly interacting sector, in which case the operator H†H can have a large anomalous
dimension that makes it effectively marginal or irrelevant.
A model along the lines of ii), conformal technicolor [39], where the strongly coupled sector is assumed to be a CFT in the UV, was in fact the motivation for the pioneering work [1]. Conformal technicolor is an interesting attempt to solve one of the long-standing problems of standard technicolor theories: how to reconcile the top mass with Flavour Changing Neutral Current (FCNC) bounds. In order to get a sizable top mass and at the same time avoid dangerous FCNCs, one has to demand that the scale dimension ∆H of
the Higgs field H is as close to one as possible. In order not to reintroduce the hierarchy 1Results for SO(N ) groups with odd N are analogous to those for SO(2N ) and do not need any special treatment.
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problem, however, one has to keep ∆H†H &4 at the same time. The analyses in refs. [1,10] have shown that generally these two conditions are in tension and that one needs ∆H &1.52
in order to have ∆H†H &4.
An alternative, phenomenologically more promising, solution is to rely on a different mechanism to generate SM fermion masses: partial compositeness [40]. To this end, one assumes that the SM fermions mix with fermion resonances of the strongly coupled sector. Due to this mixing, SM vectors and fermions become partially composite. In particular, the lighter the SM fermions are, the weaker is the mixing. This simple, yet remarkable, observation allows to significantly alleviate most flavor bounds. The Yukawa couplings are effective couplings that arise from the mixing terms once the strongly coupled states are integrated out.
This idea is particularly appealing when one assumes that the strongly coupled sector is an approximate CFT spontaneously broken at some scale µ. In this case, the hierarchy of the SM Yukawa couplings is naturally obtained by assigning different scale dimensions ∆i
ψ
to the fermion operators mixing with the different SM fermions [41]. In particular, there is no longer the need to keep ∆H close to one since the effective size of the SM Yukawa
couplings is governed by ∆i
ψ. One assumes that ∆iψ > 5/2 for all SM fermions except the
top, so that the mixing terms are irrelevant deformations of the CFT and naturally give rise to suppressed Yukawa couplings in the IR. For the top, on the other hand, one assumes that ∆t
ψ ≃ 5/2, corresponding to a nearly marginal deformation of the CFT.
One might wonder whether CFTs with all the necessary requirements to give rise to theoretically and phenomenologically viable composite Higgs models exist at all. A possible issue might arise in weakly gauging the SM subgroup of the global symmetry of the CFT. Since partial compositeness requires a fermion operator in the CFT for each SM fermion, dangerous Landau poles can potentially appear in the theory. Indeed, it has recently been shown that Landau poles represent the main obstruction in obtaining UV completions of composite Higgs models with a pNGB Higgs, based on supersymmetry [42]. It is then of primary importance to try to understand if and at what scale Landau poles will arise. In theories with a pNGB Higgs, the relevant deformation H†H can naturally be small, since it is protected by a shift symmetry. Moreover, it is not defined in the UV, where the global symmetry is restored. Nevertheless, in order not to introduce other possible fine-tunings, one should demand that any scalar operator which is not protected by any symmetry, namely which is neutral under all possible global symmetries of the CFT, should be marginal or irrelevant.
Summarizing, we can identify four properties that a CFT needs to have for a theoret-ically and phenomenologtheoret-ically viable composite Higgs model with partial compositeness:
1. A global symmetry G⊇ GSM= SU(3)c× SU(2)L× U(1)Y.
2. No scalar operator with dimension ∆ < 4 which is neutral under G.
3. No Landau poles for the SM gauge couplings below the scale ΛUV when we gauge
GSM.2
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4. The presence of fermion operators with ∆iψ ≥ 5/2 in some representation of G, such that some of its components can mix with each of the SM fermion fields. At least one fermion operator should have dimension ≃ 5/2.3
Of course, these are only necessary but not sufficient conditions to get a viable CFT. In particular, one might want to address the mechanism which gives rise to the spontaneous breaking of the conformal symmetry as well as of the global symmetry in CFTs with a pNGB Higgs.
The consistency of a CFT which fulfils the above four conditions can be checked using crossing symmetry of four-point functions of the CFT. The first and second condition can be imposed by hand, assuming the existence of the global symmetry and that the lowest-dimensional scalar operator in the singlet channel has dimension ∆S ≥ 4. One can extract
information on the third condition by analyzing the bounds on the coefficients of current-current two-point functions. Finally, the fourth condition can again be implemented by assumption. The ideal configuration would be to analyze four-point functions involving fermion operators, which by assumption should appear in the CFT, and to extract any possible information from these correlators. Although this is in principle possible to do, correlation functions involving fermions in a non-supersymmetric setting have not been worked out so far. Postponing to a future project the analysis of such correlation functions, in this paper we start to address these issues by replacing fermions with scalars with dimension 1≤ d < 2 in the third requirement.
Let us estimate how severe the Landau pole problem can be in the simplest composite Higgs model where the Higgs is the pNGB associated with the SO(5)→SO(4) symmetry breaking pattern. Let us consider the SU(3)c coupling gc, because it runs fastest and
possibly leads to the lowest-lying Landau pole, and let us denote by βCFT= gc3
κ
16π2 (2.1)
the CFT contribution to its one-loop β-function. Assuming that the only non-SM fields which are charged under SU(3)c arise from the CFT, a Landau pole develops at around
ΛL≃ µ exp 2π (κ− 7)αc(µ) (2.2) for κ > 7, where αc = gc2/(4π) and µ ∼ O(TeV) is the scale where the CFT breaks
spontaneously. Composite fermions coming from the CFT and mixing with SM fermions must be color triplets and in representations of SO(5) that give rise to electroweak SU(2) doublets and singlets. If we assume them to be in the fundamental representation 5 of SO(5), the fermion components in a given 5 can mix with both the left-handed and right-handed components of a quark field. We then need nf = 6 5s, one for each quark field, for a
total of 6×5 = 30 SU(3)ctriplet Dirac fermions. In order to have an idea of the scales which
are involved, it is useful to consider the (unrealistic) limit of a free CFT. In this case, we get κfree=
2
3× 30 = 20 , (2.3)
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corresponding to
ΛL∼ 200 TeV , (2.4)
for µ ≃ 1 TeV. It is clearly very important to set lower bounds on κ in a generic CFT, given the exponential sensitivity of ΛLon this quantity.
In the following, we will analyze bounds on the coefficients for SO(2N ) (or, equivalently, SU(N )) currents obtained from four-point functions of scalar operators in the fundamental representation of the group in presence of a gap in the operator dimension in the scalar gauge-singlet channel. In order to mimic the presence of more than one field multiplet, we will also consider fields in the bi-fundamental representation of the product group SO(2N )×SO(M).
3 Review of the bootstrap program
In this section, we briefly review the equations that one obtains by imposing crossing symmetry on four-point functions of scalar operators in a unitary CFT, and how these are numerically handled to get bounds on the CFT data. We refer the reader to ref. [1] and references therein for more details and background material.
3.1 Bootstrap equations
The four-point function of identical real scalar operators with scale dimension d in a 4D CFT can be written as hφ(x1)φ(x2)φ(x3)φ(x4)i = g(u, v) x2d 12x2d34 , (3.1) where x2ij = (xi− xj)µ(xi− xj)µ and u = x 2 12x234 x2 13x224 , v = x 2 14x223 x2 13x224 (3.2) are two conformally invariant variables. All the dynamics is encoded in the function g(u, v). Using the OPE between φ(x1)φ(x2) and φ(x3)φ(x4) (the s-channel), in the region 0≤ u, v ≤
1 this function is found to be
g(u, v) = 1 +X
∆,l
|λφφO|2g∆,l(u, v) , (3.3)
where the sum is over all primary (traceless symmetric) operators of dimension ∆ and (even) spin l that appear in the OPE and λφφO is the coefficient of the three-point function
hφφOi. The +1 in eq. (3.3) results from the contribution of the identity operator which is present in the OPE of two identical operators. The three-point functionhφφOi in this case simplifies to the two-point function hφφi which is normalized to unity. For each primary operator O, the function g∆,l(u, v), which is called a conformal block, takes into account
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and reads [43,44] g∆,l(u, v) = z ¯z z− ¯z k∆+l(z)k∆−l−2(¯z)− (z ↔ ¯z) , kβ(x)≡ xβ/22F1 β 2, β 2, β, x , (3.4)where u = z ¯z and v = (1− z)(1 − ¯z).4 Alternatively, we can obtain an expression for g(u, v) using the OPE between φ(x2)φ(x3) and φ(x1)φ(x4) (the t-channel). Demanding
that the s-channel and t-channel results for the four-point function agree gives the crossing symmetry constraint (or bootstrap equation)
X ∆,l |λO|2Fd,∆,l(z, ¯z) = 1 , (3.5) with Fd,∆,l(z, ¯z)≡ vdg∆,l(u, v)− udg∆,l(v, u) ud− vd (3.6)
and λO ≡ λφφO. When the CFT has a global symmetry G, the above analysis can be
generalized using scalar fields φa (real or complex) in some representation r of G [8].
The symmetry implies that all the field components of the multiplet must have the same dimension d. Moreover, it allows to easily classify the operators appearing in the φaφb
OPE in terms of the irreducible representations appearing in the product r⊗ r. A similar analysis applies for complex fields in the φaφ†b OPE. It is useful to introduce another
function, similar to the F of eq. (3.6): Hd,∆,l(z, ¯z)≡
vdg
∆,l(u, v) + udg∆,l(v, u)
ud+ vd . (3.7)
In presence of a global symmetry G, eq. (3.5) generalizes to a system of P + Q equations of the form X i ηpF,i X O∈ri |λOi|2Fd,∆,l(z, ¯z) = ωFp , p = 1, . . . , P , X i ηH,iq X O∈ri |λOi|2Hd,∆,l(z, ¯z) = ωHq , q = P + 1, . . . , P + Q . (3.8)
Here, i runs over all possible irreducible representations that can appear in the s- and t-channel decomposition, ηF,ip and ηqH,i are numerical factors that depend on G and λOi is a short-hand notation for thehφaφbOi three-point function coefficient. Furthermore, ωFp = 1
and ωqH =−1 if the singlet representation appears in the left-hand side of eq. (3.8), and ωFp = ωqH = 0 otherwise. The explicit form of eq. (3.8) for the cases of interest will be given in section 5and appendix B.
4Here we have used the normalization of the conformal blocks introduced in ref. [8] which differs by a factor (−2)lfrom the one of refs. [43,44].
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3.2 Bounds on OPE coefficients and numerical implementation
The bootstrap equation (3.5) has originally been used to set bounds on the scalar operator dimensions that can appear in a CFT. Shortly after that, ref. [5] has shown how to obtain bounds on the OPE coefficient λO0 of an operator O0 appearing in the φφ OPE. Let us assume that a linear functional α can be found, such that
α(Fd,∆0,l0) = 1 , α(Fd,∆,l)≥ 0 ∀(∆, l) 6= (∆0, l0) . (3.9) Applying such a functional to eq. (3.5) gives
|λO0|
2 = α(1)− X
(∆,l)6=(∆0,l0)
|λO|2α(Fd,∆,l) ≤ α(1) . (3.10)
The optimal bound is obtained by minimizing α(1) among all the functionals α which satisfy eq. (3.9). One can use the functional α also to rule out the existence of certain CFTs. For instance, if under a certain assumption on the CFT data one finds a functional α and an operator O0 for which |λO0|
2< 0, then that CFT is ruled out.
The above procedure is easily generalized in presence of global symmetries. Let us assume that we want to bound the OPE coefficient of an operatorO0with dimension ∆0and
spin l0in the representation r1. We look for a set of linear functionals αm(m = 1, . . . , P +Q)
such that P X p=1 αp ηF,1p Fd,∆0,l0 + P +Q X q=P +1 αq ηqH,1Hd,∆0,l0 = 1 , P X p=1 αp ηF,1p Fd,∆,l + P +Q X q=P +1 αq ηH,1q Hd,∆,l ≥ 0 , ∀(∆, l) 6= (∆0, l0) , P X p=1 αp ηF,ip Fd,∆,l + P +Q X q=P +1 αq ηH,iq Hd,∆,l ≥ 0 , ∀(∆, l) , i 6= 1 . (3.11)
Applying such a functional to eq. (3.8) gives
|λO0| 2 ≤ P X p=1 αp(ωFp) + P +Q X q=P +1 αq(ωHq ) . (3.12)
In our paper, we will mainly be interested in the OPE coefficient associated with a conserved vector current Jµ of a global symmetry, which has ∆0 = 3 and l0 = 1. We shall denote
this coefficient by λJ. As we will discuss in section5 (see eq. (5.9)), upper bounds on|λJ|2
turn into lower bounds on the coefficient κ introduced in eq. (2.1).
Following ref. [1], we consider functionals that act as linear combinations of derivatives on a generic function f (z, ¯z),
α(f (z, ¯z)) = X
m+n≤2k
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where amn are real coefficients. Due to the symmetries of the conformal blocks F and H,
the sum can be restricted to m < n and even values of m + n when α acts on F , and m < n and odd values of m + n when it acts on H.
We numerically search for functionals α which satisfy eqs. (3.9) and (3.11) by following the method developed in refs. [6,10]. We refer the reader to these references for further details. For this method, one approximates the derivatives of the conformal blocks Fd,∆,l
and Hd,∆,l in eq. (3.13) with polynomials Plmn(∆l(1 + x)), where x∈ [0, ∞) and ∆l = l + 2
is the unitarity bound on the scaling dimension for an operator of spin l (∆0 = 1 for
l = 0). The requirements in eq. (3.9) or eq. (3.11) imply that the linear combination of polynomials of the form amnPlmn must be positive-semidefinite on the positive real x-axis,
for any value of l. There are two great virtues in setting up the problem in this way. Firstly, there is no need to discretize the dimension ∆ and to put a cut-off value ∆max,
like in the linear programming methods used in ref. [1]. In particular, we can probe all ∆ continuously up to infinity. Secondly, one can exploit numerical packages that allow to handle very large systems of equations quite efficiently. A key variable in the numerical algorithm is the coefficient k entering in eq. (3.13). The larger k, the larger is the space of possible viable functionals, and hence the stronger are the bounds. Of course, the larger k, the more time-consuming is the numerical evaluation. For our computations, we have chosen k = 9, 10, 11, depending on the complication of the problem.
The above algorithm, however, still requires to truncate the system at a given maximal spin L. This is in principle a serious problem, because one might have
α(Fd,∆,l) < 0 for l > L . (3.14)
If L is chosen sufficiently large, O(10) or more, we do not expect possible violations in the semidefinite positiveness of α of the form (3.14) to be important for the numerical value of the bound. Indeed, large spin l implies large dimensions ∆ according to the unitarity bound, and the contribution to the four-point function of operators with large ∆ is exponentially suppressed in ∆ [30]. Nevertheless, it would be more reassuring to have more control on such effects. For parametrically large l, the conformal blocks Fd,∆,l and
Hd,∆,l and their derivatives allow for simple analytic expressions. For large l, the terms
involving the highest derivatives dominate. Using these analytic expressions, we can find the value lmax, which depends on k, for which the contribution of the large-l conformal
blocks is largest. We find, for 2k≫ 1 (see appendixA for details) lmax∼ 2k
c , (3.15)
where c =− log(12−8√2)≃ 0.377. For k ∼ 10, eq. (3.15) gives lmax∼ 50÷60. Ideally, one
would include all spins from l = 0 up to L = lmax. This is computationally quite
demand-ing. Fortunately, we have found that it is sufficient to take L = 20 to get numerically stable bounds. Changing L to L = 22 or L = 24 does not significantly alter the bounds. Neverthe-less, in order to have more control on the higher-l states, we have included two other states
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l=0 operators 1.5 2.0 2.5 3.0 3.5 4.0D
0.5 1.5 2.0 2.5 3.0 3.5Λ
0Figure 1. Upper bounds on the three-point function coefficient λ0 between two scalar operators of
dimension d = 1.6 and a scalar operatorO of dimension ∆ calculated at k = 11 with no assumptions on the spectrum (blue line) and assuming that no scalar operator in the OPE is present below ∆0= 2
(red line). For illustrative purposes, we show the free-theory value for d = 1 (in which case ∆ = 2), λfree
0 =
√
2, as a black dashed line.
in the constraints, at l = lmaxand at an intermediate value l≈ (L + lmax)/2.5 We have
nu-merically tested that this implementation works better than including states at very large values of l, such as l = 1000, 1001 as done in e.g. ref. [10]. We can always check the positivity of α a posteriori. We have found that by imposing constraints at l = 0, . . . , 20, 1000, 1001 the functional often becomes negative for values l 6= 0, . . . , 20, 1000, 1001 whereas for our implementation α remains positive for most of the l that we have checked. In practice, however, we have not detected deviations in the results among the two different implemen-tations, confirming that values of l > L are numerically negligible.
4 Bounds on OPE coefficients for tensor operators
In this section, we report our results for the upper bounds on the three-point function coefficient λOappearing in the OPE of two identical scalar operators φ of scaling dimension
d. The operator O is a traceless symmetric tensor of even spin l. The coefficient λ is normalized such that its free-theory value is
λfreeOl ≡ λ free l = √ 2 l! p(2l)!. (4.1)
We do not report the results for the l = 0 case, which were first derived in ref. [5] and subsequently improved in ref. [10]. Our results agree with figure 10 of ref. [10]. These 5More precisely, we include spins l = 35, 52 for calculations with k = 9, l = 37, 56 for k = 10 and l= 40, 60 for k = 11.
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l=2 operators 4.5 5.0 5.5 6.0 6.5 7.0 D 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Λ2 (a) l=2 operators 4.5 5.0 5.5 6.0 6.5 7.0 D 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Λ2 (b)Figure 2. Upper bounds on the three-point function coefficient λ2 between two scalar operators
of dimension d and a tensor operator O with spin l = 2 and dimension ∆ calculated at k = 11. (a) Starting from below, the lines correspond to the values d = 1.01, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6. No assumption on the spectrum is made. (b) For d = 1.62 with no assumption on the spectrum (blue line, as in (a) for d = 1.6) and assuming that no scalar operator in the OPE is present below ∆0= 2
(red line), ∆0= 3 (brown line) and ∆0= 4 (green line). For illustrative purposes, we show the
free-theory value for d = 1 (in which case ∆ = 4), λfree
2 = 1/
√
3, as a black dashed line in both panels.
bounds change if we assume that the first scalar operator which appears in the φφ OPE has a dimension ∆0 > ∆l, where ∆l is the unitarity bound on ∆. As expected, the upper
bounds do not significantly change when d is close to 1, since by continuity the theory is close to the free theory, where the only scalar operator arises exactly at ∆0 = 2. For values
of d not too close to 1, on the other hand, the bound is significantly improved and becomes more stringent as ∆0 increases. In figure 1, we report the bounds for d = 1.6 and ∆0 = 2.
Analogously, one can study the upper bounds on λ2 for generic tensor operatorsO with
spin l = 2 and dimension ∆≥ 4. Upper bounds on the central charge c ∝ 1/λ2
2 associated
with the energy-momentum tensor (the lowest-dimensional operator in the l = 2 sector, with ∆ = 4), have been extensively analyzed in refs. [6,7,10], with and without the assump-tion of a lower bound on the dimension of the lowest-lying scalar operator appearing in the φφ OPE. In figure2(a), we report the upper bounds on the coupling λ2 between two scalar
operators of dimension d and a tensor operatorO with spin l = 2 and dimension ∆ for dif-ferent d and as a function of ∆. As can be seen, the larger d is, the less stringent is the upper bound, in agreement with the naive expectation for which d− 1 can be seen as a measure (for d not too far from 1) of how strongly coupled the CFT is. Like for scalar operators, the bounds change if we make some assumptions on the CFT spectrum. As for the scalar case, the upper bounds do not significantly change when d is very close to 1, but for values of d not too close to 1, they become more stringent as ∆0 increases. For illustration, in figure2(b),
we report the upper bounds on λ2 as a function of ∆ for d = 1.62, assuming that the lowest
scalar operator appearing in the φφ OPE has a dimension ∆0 ≥ 2, ∆0≥ 3 and ∆0≥ 4.6
6The value d ≃ 1.62 is roughly the minimal one compatible with the assumption ∆
0≥4, see e.g. figure 2 of ref. [10].
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l=4 operators 6.5 7.0 7.5 8.0 8.5 9.0 D 0.1 0.2 0.3 0.4 0.5 Λ4 (a) l=4 operators 6.5 7.0 7.5 8.0 8.5 9.0 D 0.1 0.2 0.3 0.4 0.5 Λ4 (b)Figure 3. Upper bounds on the three-point function coefficient λ4 between two scalar operators
of dimension d and a tensor operator O with spin l = 4 and dimension ∆ calculated at k = 11. (a) Starting from below, the lines correspond to the values d = 1.01, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6. No assumption on the spectrum is made. (b) For d = 1.62 with no assumption on the spectrum (blue line, as in (a) for d = 1.6) and assuming that no scalar operator in the OPE is present below ∆0= 2
(red line), ∆0= 3 (brown line) and ∆0= 4 (green line). For illustrative purposes, we show the
free-theory value for d = 1 (in which case ∆ = 6), λfree
4 = 1/
√
35, as a black dashed line in both panels.
Similarly, one can analyze tensor operators at higher l. In figures 3 (a) and (b), we report the same as above for l = 4 operators. As expected, the absolute scale of λlbecomes
lower and lower as l increases, with the allowed values of λl quickly decreasing as l becomes
larger. Notice that the maximal allowed value of both λ2 and λ4 is centered at values of
∆ that increase as d is increased.
5 Bounds on current-current two-point functions
At leading order, the CFT contribution to the one-loop beta function of a gauge field Aµ, external to the CFT, is governed by the coefficient of the two point-function of the
corresponding current. Denoting by
Lgauged=LCFT+ gJAµAAµ −
1 4F
A
µνFAµν (5.1)
the total Lagrangian after the gauging, we can consider the effective action Γ(A) defined as (in euclidean signature)
e−Γ(A) = Z DΦCFT e− R d4x L gauged, (5.2)
where the functional integration is over all the CFT states and we have omitted color indices. In general
Γ(A)⊃ −1 4
Z
d4x ZFµνAFAµν, (5.3)
where Z = (1 + δZCFT) and δZCFT is the CFT contribution to the wave function
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to the RG running of g: βCFT= gµ d dµ √ Z = 1 2gµ d dµδZCFT. (5.4)By taking two functional derivatives with respect to AAµ(p) and ABν(−p) in eq. (5.2), we readily get
δABδZCFT(δµνp2− pµpν) =−g2hJµA(−p)JνB(p)ig=0, (5.5)
where the subscript in the correlator specifies that the two-point function is computed in the unperturbed CFT setting g = 0. The normalization of the current is uniquely fixed by Ward identities. Following the notation of ref. [10], we parametrize the two-point function in configuration space as follows:7
hJµA(x)JνB(0)ig=0= 3κδAB 4π4 δµν− 2 xµxν x2 1 x6 . (5.6)
The “vector central charge” κ is roughly a measure of how many charged degrees of freedom are present in the CFT, similar to the standard central charge c being a measure of the total number of degrees of freedom of the CFT. Modulo irrelevant contact terms, the momentum space correlation function reads
hJµA(−p)JνB(p)ig=0= (δµνp2− pµpν) κ 16π2δ ABlog p2 µ2 (5.7) and hence βCFT= g3 κ 16π2 . (5.8)
We extract κ by rescaling the vector current so that it appears as the coefficient of the three-point function hφiφjJµAi:
λ2J = ρ
κ. (5.9)
Upper bounds on λ2J turn into lower bounds on κ. The constant factor ρ is easily found by matching the result with the free-theory case, in which both λ2
J and κ are calculable.
In what follows, we will analyze the lower bounds on κ for different vector currents that come from the crossing symmetry constraints applied to four-point functions of scalars. 5.1 SO(N ) global symmetry
We consider a four-point function of real scalars that are taken to be the components of a single field in the fundamental representation of SO(N ) with dimension d. The crossing symmetry relations have been derived in ref. [8]. We report them here for completeness:
X S+ |λSO|2 0 F H + X T+ |λTO|2 F (1− 2 N)F −(1 + N2)H + X A− |λAO|2 −F F −H = 0 1 −1 . (5.10)
7Notice that the definition of κ here is not identical to that of ref. [10] which tacitly applies to CFTs with one charged multiplet only. In general, κhere∝PiκithereT(ri) where i runs over all the charged fields of the CFT in the representations ri and δABT(ri) = Tr(tArit
B ri).
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Here S, T and A refer respectively to the singlet, rank-2 symmetric and antisymmetric (adjoint) representations of the operators O which define the different conformal blocks. For the superscript +, only even spins are included in the sum whereas for− only odd spins are summed over. For simplicity, we have omitted the labels d, ∆, l and the arguments z, ¯z of the conformal blocks F and H. Bounds on κ (as defined in eq. (5.6)) in this set-up have already been found in ref. [10]. In this subsection we will see how these bounds change if assumptions on the dimensionality of the lowest-lying scalar operator in the singlet channel are made.
First of all, let us consider the free theory of a real scalar in the fundamental represen-tation of SO(N ) in order to fix the constant ρ in eq. (5.9). The free-theory values of the OPE coefficients in the three different channels read
λfreeA,l = √1 2λ free l (l odd) , λfreeT,l = √1 2λ free l (l even) , λfreeS,l = √1 Nλ free l (l even) , (5.11)
where λfreel is given in eq. (4.1). We in particular get λfreeJ = λfreeA,1 = 1/√2. Matching eq. (5.8) with the one-loop contribution to the β-function of a scalar in an SO(N ) gauge theory gives
κfree =
1
6, (5.12)
where we have taken T (fund.) = 1 (cf. footnote 7). From this it follows that ρ = 1/12 in eq. (5.9).
In figure 4, we report our results in terms of lower bounds on κ. We have considered the five different values N = 2, 6, 10, 14, 18 and report the lower bounds on κ for the case where no assumption on the spectrum is made (the lines starting from d = 1) and the case where the lowest-lying scalar operator in the singlet channel is assumed to have dimension ∆S≥ 4 (the other lines). The former bounds agree with previous results (e.g. compare with
figure 18 of ref. [10]). Although it is not clearly visible from the figure, we have checked that all the bounds consistently tend to the free-theory value for d→ 1. The latter bounds start from a given dcr> 1 that depends on N . This is of course expected, given the known
results for the upper bound on the dimension of the lowest-lying scalar singlet operator at a given d: CFTs at d < dcr are excluded under the assumption of a gap in the scalar
singlet sector. The values of dcr that we find agree with the values given in the literature
(compare e.g. with the dimensions d for which ∆0 = 4 in figure 4 of ref. [10]). The lower
bounds on κ become significantly more stringent when we impose that ∆S> 4. They also
decrease less rapidly when d increases compared to the unconstrained case.
In order to show how the assumption on ∆S affects the lower bounds on κ, in figure5,
we fix N = 10 and consider the three cases ∆S ≥ 2, ∆S ≥ 3 and ∆S ≥ 4. As expected, the
lower bound consistently becomes more severe as we increase ∆S. As before, the bounds
start at certain dimensions dcr which agree with previous results (compare e.g. with the
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1.2 1.4 1.6 1.8d
0.05 0.10 0.15 0.20 0.25 0.30Κ
Figure 4. Lower bounds on the two-point function coefficient κ between two conserved SO(N ) or SU(N/2) adjoint currents as obtained from a four-point function of scalar operators in the fundamental representation with dimension d calculated at k = 10. From below, the lines which start at d = 1 correspond to N = 2 (blue), N = 6 (red), N = 10 (brown), N = 14 (green), N = 18 (black), with no assumption on the spectrum. In the same order and using the same color code, the lines which start at d≃ 1.58, d ≃ 1.46, d ≃ 1.37, d ≃ 1.31 and d ≃ 1.29 show the bound which is obtained under the assumption that no scalar operator in the singlet channel has dimension ∆S < 4.
For illustrative purposes, we show the free-theory value κfree= 1/6 as a black dashed line.
5.2 SU(N ) global symmetry
We consider a four-point function of complex scalars that are taken to be the components of a field in the fundamental representation of SU(N ) with dimension d. The crossing symmetry relations have been derived in ref. [8]. We report them here for completeness:
X S± |λSO|2 F H (−)lF (−)lH 0 0 +X Ad± |λAdO |2 (1−N1)F −(1 + 1 N)H (−)l+1 1 NF (−)l+1 1 NH (−1)lF (−)lH +X T+ |λTO|2 0 0 F −H F −H +X A− |λAO|2 0 0 F −H −F H = 1 −1 1 −1 0 0 . (5.13) Here S, Ad, T and A refer respectively to the singlet, adjoint, 2 symmetric and rank-2 antisymmetric representations of the operators O which define the different conformal blocks. For the superscript +, even spins are included in the sum, and for−, odd spins are summed over. We consider here the lower bounds on κ (as defined in eq. (5.6)) associated with the adjoint current.
As in subsection 5.1, we start by looking at the free theory of a complex scalar in the fundamental representation of SU(N ) in order to fix the constant ρ in eq. (5.9). The
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1.2 1.4 1.6 1.8d
0.05 0.10 0.15 0.20 0.25Κ
Figure 5. Lower bounds on the two-point function coefficient κ between two conserved SO(10) or SU(5) adjoint currents as obtained from a four-point function of scalar operators in the fundamental representation with dimension d calculated at k = 10. From below, the lines correspond to the case with no assumption on the spectrum (blue) and assuming that no scalar operator in the singlet channel has dimension ∆S < 2 (red), ∆S < 3 (brown), ∆S < 4 (green). For illustrative purposes,
we show the free-theory value κfree= 1/6 as a black dashed line.
free-theory values of the OPE coefficients in the four different channels read λfreeAd,l = √1
2λ
free
l (l even and odd) ,
λfreeS,l = √1 2Nλ
free
l (l even and odd) ,
λfreeT,l = √1 2λ free l (l even) , λfreeA,l = √1 2λ free l (l odd) , (5.14)
where λfreel is given in eq. (4.1). We in particular get λfreeJ = λfreeAd,1 = 1/√2. Matching eq. (5.8) with the one-loop contribution to the β-function of a complex scalar in an SU(N ) gauge theory gives
κfree =
1
6, (5.15)
where we have taken T (fund.) = 1/2 (cf. footnote 7). From this it follows that ρ = 1/12 in eq. (5.9) as for SO(N ).
The six crossing symmetry equations (5.13) should reduce to the three equations (5.10) when the group SU(N ) is embedded in an underlying SO(2N ) group. The decomposition of the singlet, adjoint and rank-2 symmetric representations of SO(2N ) in terms of SU(N )
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representations reads
SSO(2N )+ = SSU(N )+ ,
TSO(2N )+ = TSU(N )+ ⊕ T+SU(N )⊕ Ad+SU(N ),
A−SO(2N )= A−SU(N )⊕ A−SU(N )⊕ Ad−SU(N )⊕ SSU(N )− .
(5.16)
If SU(N )⊂ SO(2N), for each primary operator in the A− (T+) representation of SU(N ),
there is a corresponding operator in the Ad−and S−(Ad+) representation as follows from
eq. (5.16). The OPE coefficients of these operators are related by the underlying SO(2N ) symmetry, λTSO(2N )+ = λAdSU(N )+ , λSO(2N )A− = λASU(N )− = √N λSSU(N )− . It is straightforward to check with these identifications that eqs. (5.13) reduce to eqs. (5.10).
As we have already mentioned, the numerical results for the lower bounds on κ for SU(N ) are identical to those for SO(2N ), see figure 4. This suggests that, given a set of three functionals αm that satisfy eq. (3.11) with P = 2, Q = 1 and ηF,H as given by
eq. (5.10), one should be able to construct a set of six functionals ˜αm as linear combinations
of the αmsuch that these functionals satisfy eq. (3.11) with P = 3, Q = 3 and ηF,H as given
by eq. (5.13). It would be interesting to find such a mapping and hence to understand in more analytical terms why the bounds on κ for SO(2N ) and SU(N ) are equal.
5.3 G1 × G2 global symmetries
The lower bounds on κ found in subsections5.1and5.2apply to CFTs in presence of at least one scalar field in the fundamental representation of G1, where G1= SO(M ) or SU(M ). Of
course, the CFT can contain additional charged fields, for example a number N of scalars in the fundamental representation of G1, with dimensions d1, . . . dN. In the free-theory
limit of N decoupled scalars (real for SO(M ), complex for SU(M )) we would simply have κfree=
N
6 . (5.17)
The larger N is, the more constraining (and interesting) the lower bounds are. One cannot naively rescale the results of figure 4 by a factor of N in order to match the new free theory limit, however, because the interactions among the scalars will not be taken into account in this way. A more constraining bound could likely be obtained by studying the coupled set of four-point functions involving all N scalars. This is in general not straightforward to do, since the crossing symmetry constraints are significantly more involved in presence of fields with different scaling dimensions.8 A simple way to mimic the presence of more fields charged under a given group, though at the cost of assuming identical scaling dimensions d1 = . . . dN = d, is obtained by introducing a further global
symmetry group G2 and assuming that the N fields transform under some representation
of G2. This is the main motivation for us to consider global symmetries which are direct
products of two simple groups: it is a way to obtain lower bounds on κG1 in presence of more than one field charged under G1. More specifically, in the following we will consider
fields in the fundamental representation of G2 = SO(N ).
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1.2 1.4 1.6 1.8d
0.0 0.5 1.0 1.5Κ
Figure 6. Lower bounds on the two-point function coefficient κ between two conserved SO(N ) (or SU(N/2)) adjoint currents as obtained from a four-point function of scalar operators with dimension d in the bi-fundamental representation of SO(N )×SO(M), calculated at k = 9. We take N = 6. From below, the lines which start at d = 1 correspond to M = 2 (blue), M = 6 (red), M = 10 (brown), with no assumption on the spectrum. In the same order and using the same color code, the lines which start at d≃ 1.34, d ≃ 1.28 and d ≃ 1.25 show the bound which is obtained under the assumption that no scalar operator in the singlet channel has dimension ∆S < 4.
5.3.1 SO(N )×SO(M )
Consider a CFT with global symmetry SO(N )×SO(M) and one real scalar φia in the bi-fundamental representation of SO(N )×SO(M), a and i being SO(N) and SO(M) indices, respectively. In complete analogy to the SO(M ) case discussed in ref. [8], we can impose crossing symmetry in the s- and t-channel on the four-point function hφi
a(x1)φjb(x2)φck(x3)φld(x4)i in order to obtain the bootstrap equations. The operators
appearing in the φφ OPE transform under SO(N )×SO(M) according to the decompo-sition of (N, M)⊗ (N, M), where N and M denote the fundamental representations of respectively SO(N ) and SO(M ). This gives 9 different representations, consisting of pairs (ij), where i, j = S, T, A refer to the singlet (S), symmetric (T ) and antisymmetric (A) representations of respectively SO(N ) and SO(M ). Correspondingly, we get a total of 3× 3 = 9 equations. We report them in eq. (B.2) in appendix B.1.
The SO(N ) conserved current that we analyze is in the (AS) representation and is the lowest-dimensional operator appearing in the functions FAS and HAS defined in eq. (B.1).
In figure6, we show the lower bounds on κSO(6)for the three cases M = 2, 6, 10. While for the case of SO(N ) or SU(N/2) considered before the lower bound first becomes significantly more stringent with growing d and only from a certain d onwards becomes less stringent, here a slight increase in the bound arises only for d very close to 1 after which the bound decreases with d. The lines starting from d = 1 correspond to the case where no assumption
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on the spectrum has been made, while for the other lines we have assumed that the lowest-lying scalar operator in the SS channel has dimension ∆SS ≥ 4. The lower bounds for the
latter case are stronger than for the former, but the difference is less substantial than for the groups SO(N ) or SU(N/2). The lower bound d≥ dcr on the dimension of φia above which
the lowest-lying operator in the SS channel can have a dimension ∆SS ≥ 4 is also weaker
than what was found for SO(N ) or SU(N/2). This is expected, since this bound becomes the weaker the larger the group is. The correct free-theory limit is obtained in all three cases. The shape of the lower bound on κ with no assumption on the spectrum in figure 6
resembles the bound found in ref. [10] for SU(N ) singlet currents (see e.g. their figure 19). From the SO(M ) point of view, the SO(N ) current is in fact a collection of N (N−1)/2 sin-glet currents. On the other hand, for N ≫ M, we find that the lower bound on κ for SO(N) currents shows the characteristic bump of single SO(N ) or SU(N/2) currents, well above the free-theory value, as in figure4 (a). For illustration, we show the bound on κ obtained for N = 30 and M = 2 in figure7. It would be interesting to further explore these bounds and to understand the origin of their different behaviours in the regimes N ≤ M and N ≫ M. As a final application of our results, in figure8, we report the lower bounds on κ for the group SO(6)×SO(120). We choose SO(120) because the contribution of 120 free complex scalar triplets to the SU(3)c ⊂ SO(6) current-current two-point function gives κ = 20.
This in turn is the same value found in eq. (2.3) for the number of free fermion triplets which are needed to give mass to all the SM quarks in the SO(5)→SO(4) pNGB composite Higgs model mentioned in section2. We consider SO(6)×SO(120) and not SU(3)×SO(120) because the latter case is computationally very demanding (incidentally, in one of the models presented in ref. [42], SU(3)cwas actually embedded in an underlying SO(6) flavour
global symmetry). Anyhow, given the equivalence between the SO(2N ) and SU(N ) lower bounds on κ, we believe that these results would also hold for the SU(3)×SO(120) case.
As we see in figure 8, assuming the absence of a relevant scalar singlet operator in the CFT does not significantly change the bounds. Furthermore, the most dangerous region regarding Landau poles, which is the region close to d = 1, is not consistent with the assumption of absence of relevant deformations. If we demand that no sub-Planckian Landau pole arises, then we need d & 1.2, while for d & 1.25, αcremains asymptotically free.
5.3.2 SO(N )×SU(M )
Consider a CFT with global symmetry SO(N )×SU(M) and one complex scalar φia in the bi-fundamental representation of SO(N )×SU(M), a and i being SO(N) and SU(M ) indices, respectively. We impose crossing symmetry in the s- and t-channel on the four-point function hφia(x1)φ
¯j†
b (x2)φkc(x3)φ ¯ l†
d(x4)i and the four-point function with
x3 ↔ x4. The operators appearing in the φφ OPE transform under SO(N )×SU(M)
in representations (ij), where i = S, T, A refer to the singlet (S), symmetric (T ) and antisymmetric (A) representations of SO(N ) and j = A, T refer to the symmetric (T ) and antisymmetric (A) representations of SU(M ). This gives 6 different representations, with even and/or odd spin operators, depending on the representation. The operators appearing in the φφ† OPE transform in representations (ij), with i = S, T, A as before,
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1.2 1.4 1.6 1.8d
0.0 0.1 0.2 0.3 0.4 0.5Κ
Figure 7. Lower bounds on the two-point function coefficient κ between two conserved SO(30) (or SU(15)) adjoint currents as obtained from a four-point function of scalar operators with dimension d in the bi-fundamental representation of SO(30)×SO(2), calculated at k = 9. No assumption on the spectrum is made. The black dashed line corresponds to the free-theory value κfree= 1/3.
1.0 1.2 1.4 1.6 1.8 2.0
d
0 5 10 15 20Κ
Figure 8. Lower bounds on the two-point function coefficient κ between two conserved SO(6)⊃ SU(3)cadjoint currents as obtained from a four-point function of scalar operators with dimension d
in the bi-fundamental representation of SO(6)×SO(120), calculated at k = 9. The line which starts at d = 1 corresponds to the case where no assumption is made on the spectrum, whereas for the line which starts at d≃ 1.19, the CFT is assumed to have no scalar operator in the singlet channel with dimension ∆S < 4. In the green region αc remains asymptotically free, while in the orange
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Taking into account that SU(M ) singlet and adjoint operators appear with both even and odd spins, we get 12 different conformal blocks, for a total of 18 bootstrap equations. We report them in eqs. (B.5) and (B.6) in appendix B.2. The SO(N ) conserved current that we are interested in transforms under the AS representation and is the lowest-dimensional operator appearing in the functions FAS− and HAS− defined in eq. (B.4).
The large number of bootstrap equations makes the numerical analysis of the SO(N )×SU(M) case computationally very demanding. We have however been able to compute the lower bounds on κSU (M ) at level k = 6 with L = 15, using N = 30 and M = 3 and either assuming that no scalar operator has dimension ∆S < 4 or without any
assumption. The results are essentially indistinguishable from those obtained for κSO(6) when considering SO(N )×SO(2M), though the accuracy obtained is not enough to claim that they are identical. Given also the observed equivalence of the bounds for SU(M ) and SO(2M ), we take this result as enough evidence to conjecture that the bounds for SO(N )×SU(M) are identical to those for SO(N)×SO(2M), for any N and M.
6 Conclusions
In this paper, we have numerically studied bounds on various OPE coefficients in 4D CFTs using the bootstrap approach applied to four-point functions of scalar operators. We have first studied bounds on OPE coefficients of symmetric traceless tensor operators with spins l = 2 and l = 4 as a function of their scaling dimension. Furthermore, we have analyzed how an assumption on the dimension of the lowest-lying scalar operator affects such bounds.
We have then considered 4D CFTs with a global symmetry G. When this group, or a subgroup of it, is gauged by weakly coupling external gauge fields to the CFT, the coefficient κ which enters in the two-point function of the associated conserved vector currents governs the leading CFT contribution to the one-loop β-function of the corresponding gauge cou-pling. In particular, if this contribution is too large, it gives rise to unwanted sub-Planckian Landau poles. Motivated by physics beyond the Standard Model, where GSM⊆ G, we have
numerically studied the lower bounds on the coefficient κ using techniques developed in ref. [10]. Possible hierarchy problems are avoided by demanding that all scalar operators in the spectrum which are singlets under the global symmetry have dimensions ∆S ≥ 4.
More specifically, we have considered lower bounds on κ extracted from crossing symmetry constraints on four-point functions of scalar operators φi in the fundamental representation
of SO(N ), or the bi-fundamental representation of SO(N )×SO(M).
We have been mostly motivated by applications in the context of composite Higgs models with partial compositeness, where the CFT is assumed to have a global symmetry G and a set of fermion operators with different dimensions. Our results are encouraging. For concreteness, we have considered a CFT with SU(3)c ⊂ SO(6) global symmetry. We
have chosen the scalar matter content of the CFT such that, in the free-theory limit, it has the same quantitative effect on the β-function of αc as the fermion matter content in
a popular composite pNGB Higgs model based on the coset SO(5)/SO(4). In this setting, Landau poles for αc can always be avoided as long as the dimension of the external scalar
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Of course, the bootstrap constraints that we have obtained are only lower bounds, and it is well possible that these bounds can be significantly improved. For example, imposing the bootstrap equations for several field multiplets directly without invoking an additional symmetry, along the lines of ref. [45], might lead to stronger bounds compared to the rather weak bounds on SO(N) currents that we have found for CFTs with SO(N)×SO(M) global symmetry when N ≤ M. Our results are obviously not directly applicable to the actual phenomenological models, because i) we have considered four-point functions of scalars and not fermions and ii) we have assumed that the scalars all have the same scaling dimension. Nevertheless, we believe that these preliminary results can be a useful first step towards a more comprehensive analysis that goes beyond these two limitations.
Acknowledgments
We thank Alfredo Urbano, Balt van Rees, David Poland and especially Slava Rychkov for useful discussions and correspondence. We are particularly grateful to the INFN technical staff of the Zefiro cluster in Pisa, where most of the computations for this paper have been performed. B.v.H. and M.S. thank the Galileo Galilei Institute for Theoretical Physics for hospitality during the completion of this work.
A Details about the numerical procedure
In order to discuss the numerical procedure developed in ref. [10], let us consider the simplest case of an external singlet operator. The constraints that the functional α needs to fulfill in order to get bounds on OPE coefficients are given in eq. (3.9). It is convenient to first rescale the bootstrap equation by a (∆, l)-independent function f (z, ¯z) (see ref. [10] for more details). In particular, the positivity constraints on the rescaled conformal blocks Ed,∆,l+ ≡ f(z, ¯z)Fd,∆,l then read
amn∂zm∂zn¯Ed,∆,l+ ≥ 0 ∀(∆, l) 6= (∆0, l0) , (A.1)
where summation over m and n is understood and the derivatives are evaluated at z = ¯z = 1/2. The crucial insight is that the derivatives of Ed,∆,l+ allow for an approximation
∂zm∂zn¯Ed,∆,l+ ≃ χl(∆)Ul,d,+mn (∆) , (A.2)
where χl(∆) is a positive definite function of ∆ and Ul,d,+mn (∆) is a polynomial in ∆. We use
5 roots for this approximation (see ref. [10] for more details). An analogous approxima-tion can be found for the rescaled conformal blocks Ul,d,−mn (∆)≡ ˜f (z, ¯z)Hd,∆,l that appear
when dealing with global symmetries. Making use of a theorem by Hilbert, the positivity constraints in eq. (A.1) can equivalently be formulated as the requirement that there exist positive semidefinite matrices Al and Bl such that
amnUl,d,+mn (∆l(1 + x)) = XpAlXpT + x XqBlXqT ∀l 6= l0. (A.3)
Here Xp ≡ (1, x, . . . , xp) is a vector and p and q are determined by the degree of the
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The task now consists of finding coefficients amn and a set of matrices Al and Bl such
that eq. (A.3) is fulfilled (additional constraints arise from e.g. the normalization condition α(Fd,∆0,l0) = 1 in eq. (3.9) and the minimization of α(1) in eq. (3.10)). This can be formulated as a positive semidefinite program for which there exist powerful numerical codes. The existence of such coefficients and matrices guarantees the positivity of the functional for all ∆≥ ∆l (corresponding to x≥ 0). As already discussed in section3.2, on
the other hand, we can only take a finite number of spins l into account.
We use Mathematica 9.0 to calculate the coefficients of the polynomials Ul,d,±mn and to set up the positive semidefinite program. The data is written to file and handed to the numerical code SDPA-GMP 7.1.2 [46] (using their sparse data format) which solves the positive semidefinite program. We use the same parameter set for the SDPA-GMP as ref. [10] (see the table in their appendix B). For the calculations, we have used the Zefiro cluster of the INFN which is located in Pisa (Italy). This cluster consists of 25 computers, each of which has 512 GB RAM and 4 processors with 16 cores. For the plots, we have calculated points with a spacing of δd = 3· 10−2 or δ∆ = 3· 10−2. In order to obtain
smooth plots, we interpolate between these points.
The Multiple Precision Arithmetic Library (GMP) allows to carry out calculations up to high precision. This is necessary because the numerical values of the coefficients of the polynomials Ul,d,±mn span several orders of magnitudes. An important source for this spread are conformal blocks with large spins l & 10. For these values of l, an asymptotic expression for the conformal blocks and its derivatives is a good approximation. Taking z = 1/2 + a + b and ¯z = 1/2 + a− b, for l2 ≫ ∆ − l − 2 one finds [1]
∂a2m∂b2nFd,∆,l|a=b=0 ≃
const.
(2m + 1)(2n + 1)(2 √
2l)2m+2n+2e−c l, (A.4) where c =− log(12 − 8√2)≃ 0.377 and const. is a positive constant of O(1) that only de-pends on d. A straightforward generalization of this result allows to also find an asymptotic analytic expression for the conformal block H defined in eq. (3.7):
∂a2m∂b2nHd,∆,l|a=b=0 ≃
const. (2n + 1)(2
√
2l)2m+2n+1e−c l. (A.5) From the above two results, we find that the spread among the coefficients of the polynomi-als for a given spin l & 10 is at least of orderO(l2k+2) for conformal blocks F andO(l2k+1)
for H. In addition, these results allow us to estimate the value lmaxfor which derivatives of
the conformal blocks have a maximum (in which case potential violations of the positivity constraint in eq. (3.9) could give a large correction in eq. (3.10)). To this end, notice that for a given l, the largest coefficients arise from the highest derivatives with m + n < 2k. Maximizing these coefficients with respect to l then yields the formula for lmaxin eq. (3.15).
An additional source for the spread arises from the approximation in eq. (A.2). The functions χl(∆) are numerically small for large spins l and therefore increase the spread
among the various coefficients of the polynomials Umn
l,d,±(∆) ≃ ∂zm∂zn¯Ed,∆,l± / χl(∆) that
determine the positive semidefinite program.
In order to reduce the numerical spread among the polynom coefficients (which allows to reduce the required precision and thereby speeds up the calculation), we rescale them
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by both an (m, n)-dependent factor and an l-dependent factor before handing them to the SDPA-GMP. Both of these rescalings transform the positivity constraint eq. (A.3) into an equivalent constraint. Indeed, the (m, n)-dependent factor amounts to a redefinition of the coefficients amn, whereas the l-dependent rescaling can be absorbed into the matrices Al
and Bl. Note, however, that e.g. the effect of the rescaling on the normalization condition
α(Fd,∆0,l0) = 1 in eq. (3.9) needs to be taken into account when calculating the bound from eq. (3.10).
B Crossing relations for SO(N )×SO(M ) and SO(N )×SU(M )
We report here the crossing symmetry constraints coming from four-point functions of scalar operators with scaling dimensions d in the bi-fundamental representation of SO(N )×SO(M) and SU(N)×SO(M).
B.1 SO(N )×SO(M )
Let φi
a be the scalar operator in the bi-fundamental representation of SO(N )×SO(M),
with a and i being SO(N ) and SO(M ) indices, respectively. As usual, we define conformal blocks that contain the contributions of the operators appearing in the OPE of φiaφjb in a given representation of the global symmetry. We have nine different conformal blocks Gij, where i, j = S, T, A with S, T and A corresponding to the singlet, symmetric and
antisymmetric representations of SO(N ) and SO(M ). The first index refers to SO(N ), the second one to SO(M ). The spin of the operators entering in Gij is even if zero or
two antisymmetric representations appear and odd otherwise. In order to have reasonably compact formulas, we define the functions
Fij ≡ X O∈(i,j)−sector |λijO|2 Fd,∆,l(z, ¯z) , Hij ≡ X O∈(i,j)−sector |λijO|2 Hd,∆,l(z, ¯z) . (B.1)
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In terms of these, the crossing relations read FSS − 2 MFST − 2 NFT S+ 1 + 4 M N FT T + FAT + FT A+ FAA= 1 , HSS− 2 MHST − 2 NHT S− 1− 4 M N HT T − HAT − HT A− HAA=−1 , 1− 2 M FT T+ FT S− FAS+ FT A− 1− 2 M FAT − FAA= 0 , 1 + 2 M HT T − HT S+ HAS+ HT A− 1 + 2 M HAT − HAA= 0 , 1− 2 N FT T+ FST − FSA+ FAT − 1− 2 N FT A− FAA= 0 , 1 + 2 N HT T − HST + HSA+ HAT − 1 + 2 N HT A− HAA= 0 , 2 M + 2 N FT T + 2 NFT A+ 2 MFAT − FT S− FST − FSA− FAS = 0 , 2 M − 2 N HT T − 2 NHT A+ 2 MHAT − HT S+ HST + HSA− HAS = 0 , FT T − FAT − FT A+ FAA= 0 . (B.2)
We have verified that reflection positivity is satisfied in the appropriate channels. The values of the OPE coefficients in the free-theory limit d→ 1 read
λT Tl = λAAl = λATl = λT Al = 1 2λ free l , λT Sl = λASl = √1 2Mλ free l , λSTl = λSAl = √1 2Nλ free l , λSSl = 1 2√M Nλ free l , (B.3)
where λfreel is given in eq. (4.1) and l is even or odd depending on the representation. Consistency with the free-theory limit provides a further check on various signs appearing in eq. (B.2).
B.2 SO(N )×SU(M )
Let φiaand φ¯i,†a be a scalar operator and its complex conjugate in the bi-fundamental
repre-sentation of SO(N )×SU(M), with a and i being SO(N) and SU(M) indices, respectively. As usual, we define conformal blocks that contain the contributions of the operators appear-ing in the OPE of φiaφjb in a given representation of the global symmetry. Since operators in the singlet and adjoint representations of SU(M ) can have both even and odd spin, we define
Fij+/−≡ X O∈(i,j)−sector l even/odd |λij+/− O |2 Fd,∆,l(z, ¯z) , H +/− ij ≡ X O∈(i,j)−sector l even/odd |λij+/− O |2 Hd,∆,l(z, ¯z) , Fij ≡ Fij++ Fij−, Fˆij ≡ Fij+− Fij−, Hij ≡ Hij++ Hij−, Hˆij ≡ Hij+− Hij−. (B.4)
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Here i runs over the representations S, T, A of SO(N ), while j runs over the singlet (S), adjoint (Ad), symmetric (T ) and antisymmetric (A) representations of SU(M ). Distinguishing between even and odd spins, we have a total of 18 conformal blocks and, correspondingly, a system of 18 crossing symmetry constraints. Six of these constraints arise by imposing crossing symmetry in the s- and t-channel on the four-point function hφi aφ ¯ j,† b φkcφ ¯ l,† d i. They read FSS− 2 NFT S− 1 MFSAd+ 1 + 2 M N FT Ad+ FAAd= 1 , HSS− 2 NHT S− 1 MHSAd− 1− 2 M N HT Ad− HAAd=−1 , FT S− FAS + 1− 1 M FT Ad− 1− 1 M FAAd= 0 , HT S− HAS− 1 + 1 M HT Ad+ 1 + 1 M HAAd= 0 , FT S+ FAS − 1 M + 2 N FT Ad+ FSAd− 1 MFAAd= 0 , HT S+ HAS+ −M1 + 2 N HT Ad− HSAd− 1 MHAAd= 0 . (B.5)
The remaining twelve constraints arise by imposing crossing symmetry in the s- and t-channel on the four-point function hφiaφ¯j,†b φ¯k,†c φldi. They read
ˆ FSS− 2 NFˆT S− 1 MFˆSAd+ 2 M NFˆT Ad+ F + T T + F − AT + F − T A+ FAA+ = 1 , ˆ HSS− 2 NHˆT S− 1 MHˆSAd+ 2 M NHˆT Ad− H + T T − H − AT − H − T A− HAA+ =−1 , ˆ FT S− 1 MFˆT Ad− 1 MFˆAAd+ ˆFAS+ F + T T − F − AT + F − T A− FAA+ = 0 , ˆ HT S− 1 MHˆT Ad− 1 MHˆAAd+ ˆHAS− H + T T + HAT− − HT A− + HAA+ = 0 , ˆ FT S− 1 MFˆT Ad+ 1 MFˆAAd− ˆFAS− 2 NF + T T + FST+ + FSA− − 2 NF − T A= 0 , ˆ HT S− 1 MHˆT Ad+ 1 MHˆAAd− ˆHAS+ 2 NH + T T− HST+ − H − SA+ 2 NH − T A= 0 , ˆ FSAd− 2 NFˆT Ad+ F + T T − F − T A+ F − AT − FAA+ = 0 , ˆ HSAd− 2 NHˆT Ad− H + T T + H − T A− H − AT + HAA+ = 0 , ˆ FT Ad+ ˆFAAd+ FT T+ − FT A− − FAT− + FAA+ = 0 , ˆ HT Ad+ ˆHAAd− HT T+ + HT A− + HAT− − HAA+ = 0 , ˆ FT Ad− ˆFAAd− 2 NF + T T + FST+ − F − SA+ 2 NF − T A= 0 , ˆ HT Ad− ˆHAAd+ 2 NH + T T− H + ST + H − SA− 2 NH − T A= 0 . (B.6)
Reflection positivity fixes the signs in both the s- and t-channel for hφi aφ ¯ j,† b φkcφ ¯ l,† d i. By
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c↔ d, this then also fixes the signs in the s-channel (the channel for which the φφ† OPE
is used) forhφiaφ¯j,†b φ¯k,†c φldi. The signs in the t-channel for the latter four-point function are
in turn fixed by reflection positivity. The values of the OPE coefficients in the free-theory limit d→ 1 read λT Ad+ l = λ T Ad− l = λ AAd+ l = λ AAd− l = λ T T+ l = λ T A− l = λ AT− l = λ AA+ l = 1 2λ free l , λSAd+ l = λ SAd− l = λ ST+ l = λ SA− l = 1 √ 2Nλ free l , λT S+ l = λ T S− l = λ AS+ l = λ AS− l = 1 2√Mλ free l , λSS+ l = λ SS− l = 1 √ 2M Nλ free l , (B.7)
where λfreel is given in eq. (4.1) and l is even or odd depending on the representation. Consistency with the free-theory limit provides a further check on various signs appearing in eqs. (B.5) and (B.6). As a further consistency check, we have verified that eqs. (B.5) and (B.6) reduce to eqs. (B.2) when SO(N )× SU(M) ⊂ SO(N) × SO(2M).
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